Question

Consider the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cylinder \((x-1)^{2}+y^{2}=1,\) for \(z \geq 0\) a. Find the surface area of the cylinder inside the sphere. b. Find the surface area of the sphere inside the cylinder.

Short answer

Question: Calculate the surface areas of (a) the cylinder inside the sphere, and (b) the sphere inside the cylinder. Answer: a. The surface area of the cylinder inside the sphere is \(A_{cylinder} = 4z \times \int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\) b. The surface area of the sphere inside the cylinder is \(A_{sphere} = 4 \times \int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\)

Step-by-step solution

Find the intersection points between the sphere and the cylinder.

We have the equations for the sphere: \(x^{2}+y^{2}+z^{2}=4\) and the cylinder: \((x-1)^{2}+y^{2}=1\). To find the intersection points, we can substitute the equation for the cylinder into the equation for the sphere and solve for x, y, and z. To do this, first expand the equation for the cylinder: \((x-1)^2 + y^2 = 1 \Rightarrow x^2 - 2x +1 + y^2 =1\). Then substitute the equation for the sphere, replacing x^2 and y^2: \(x^2 + y^2 + z^2 = x^2 - 2x + 1 + y^2 = 4 \Rightarrow z^2 = 2x + 3\) Now, substitute the equation for the cylinder back into x^2 + y^2: \(x^2 + y^2 = 4 - z^2 = 4 - (2x + 3) \Rightarrow x^2 - 2x - y^2 + 1 = 0\) Now, we need to solve for one variable (x, y, or z) using the equations we have derived. In this case, we can rewrite the equation of the cylinder as \(y^{2} = 1 - (x-1)^{2}\) and after finding the value of z, we can substitute it back into the surface area calculations for the sphere and the cylinder.

Find the arc length for the intersection points between the sphere and the cylinder.

The intersection points form a curve along which the two shapes intersect. To find the arc length, we need to find the integral of the curve. We can parametrize the curve with a parameter t: \(\bold{r}(t) = (x(t), y(t), z(t)) = (1 + \cos{t}, \sin{t}, z(t))\) As the intersection points are symmetric with respect to the xy-plane, we can find the arc length for half of the curve and then double the result. To find the arc length, we first need to find the derivative of \(\bold{r}(t)\): \(\frac{d\bold{r}(t)}{dt} = (-\sin{t}, \cos{t}, \frac{dz}{dt})\) The length of the derivative is given by: \(|\frac{d\bold{r}(t)}{dt}| = \sqrt{(-\sin{t})^2 + (\cos{t})^2 + (\frac{dz}{dt})^2} = \sqrt{1 + (\frac{dz}{dt})^2}\) To find the arc length, we need to integrate the length of the derivative over the interval \([0, \pi]\): \(L = 2 \int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\)

Find the surface area of the cylinder inside the sphere.

The surface area of the cylinder can be found by multiplying the height with the arc length. First, find the height; based on symmetry, the height of the cylinder inside the sphere is equal to \(2z\). Finally, find the surface area of the cylinder: \(A_{cylinder} = \text{height} \times L = 2z \times 2\int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\)

Find the surface area of the sphere inside the cylinder.

The spherical surface area corresponding to the intersection curve can be found by multiplying the radius of the sphere with the arc length. The sphere's radius is given as \(r = 2\). Finally, find the surface area of the sphere inside the cylinder: \(A_{sphere} = r \times L = 2 \times 2\int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\) a. The surface area of the cylinder inside the sphere is \(A_{cylinder} = 4z \times \int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\) b. The surface area of the sphere inside the cylinder is \(A_{sphere} = 4 \times \int_{0}^{\pi} \sqrt{1 + (\frac{dz}{dt})^2} dt\)

Key concepts

Intersection Points of Sphere and Cylinder

Understanding the intersection points of a sphere and a cylinder is crucial in many areas of geometry and calculus. Consider the sphere with the equation \(x^{2}+y^{2}+z^{2}=4\) and the cylinder given by \( (x-1)^{2}+y^{2}=1 \) for \( z \geq 0 \). To determine where these two surfaces intersect, we substitute one equation into the other and solve for the variables. This process often involves expanding and rearranging terms to isolate one of the variables.

Expanding the cylinder’s equation yields \( x^2 - 2x +1 + y^2 =1\), which we then equate with the sphere's to find \( z^2 = 2x + 3\). By using substitution, we can express \( y^2 \) in terms of \( x \) and solve for \( x \) or \( y \) directly, eventually finding the intersecting points that form a curve on the surfaces of the shapes. This curve is the key ingredient for calculating the arc length around the intersection and later computing the surface areas within their intersection.

Arc Length Calculation

The concept of arc length is essential in understanding the geometry of curves. When two surfaces, such as a sphere and a cylinder, intersect, they form a curve along their intersection. To calculate the arc length, we must integrate along this curve.

For ease of calculation, this curve can be represented using parametric equations, and the arc length is the definite integral of the derivative of these parametric functions over the parameter's interval. Specifically, in our example with \(\bold{r}(t) = (1 + \cos{t}, \sin{t}, z(t))\), we find the derivative with respect to \( t \) and then the magnitude of this derivative vector to get the integrand for the arc length. We then integrate this over the interval \( [0, \pi] \) because the intersection points are symmetrical about the \( xy-plane \).

The actual calculation involves determining \( \left|\frac{d\bold{r}(t)}{dt}\right| \) and requires integrating \( \sqrt{1 + (\frac{dz}{dt})^2} \) between 0 and \( \pi \). Because of the symmetry, we can calculate for half the curve and double the result to get the total arc length around the intersecting curve of the sphere and cylinder – a fundamental step to finding the surfaces' respective areas within this intersection.

Parametric Equations

Parametric equations are a powerful tool used to describe curves and surfaces in a multi-dimensional space. These equations express the coordinates of the points on a curve as functions of a single parameter, typically \( t \). In the context of finding the arc length of the curve at the intersection of a sphere and a cylinder, we use parametric equations to succinctly represent the coordinates of any point along the curve.

In our example, the intersection curve is represented parametrically as \(\bold{r}(t) = (1 + \cos{t}, \sin{t}, z(t))\). The methods for finding arc length require that we take the derivative of these parametric equations with respect to \( t \) to determine the rate of change and speed of a point moving along the curve. This is integral (no pun intended) to computing the arc length because it provides the basis for the integrand used in the arc length formula.

By understanding parametric equations, students gain insights into how complex curves can be tackled and arc lengths can be found, ultimately leading to determining the surface areas of complex shapes. This process exemplifies the combined application of concepts in geometry, algebra, calculus, and trigonometry in problem-solving.